Some History

There’s an interesting new preprint by the historian of mathematics Erhard Scholz about the early history of the use of representation theory in quantum mechanics. Immediately after the beginnings of quantum mechanics in 1925, several people started to realize that the representation theory of the symmetric and rotation groups was a very powerful tool for getting at some of the implications of quantum mechanics for atomic spectra. One of the main figures in this was Eugene Wigner, who was trained as a chemical engineer, but worked on this topic with his fellow Hungarian, the well-known mathematician von Neumann.

Equally important was the role of the mathematician Hermann Weyl, who in 1925 had just completed his main work on the representation theory of compact groups, perhaps the most important mathematical work in a very illustrious career. Weyl was in close communication both with the group at Gottingen (Heisenberg, Born, Jordan) who were developing matrix mechanics, as well as Schrodinger who was working on wave mechanics. Weyl and Schrodinger both were professors in Zurich and knew each other well (Schrodinger’s first paper on quantum mechanics thanks Weyl for explaining to him some of the general properties of equations such as the Schrodinger equation). In 1927/8 Weyl gave a course on quantum mechanics and representation theory, which became the basis of his extremely influential book “The Theory of Groups and Quantum Mechanics”, first published in 1928.

Scholz has also posted another preprint about Weyl’s work, one that focuses on how his conception of the relation between matter and geometry evolved from 1915 to 1930. Weyl worked on general relativity and wrote an influential book about it (Space-Time-Matter, 1918). At that time he, Einstein and others believed that matter could somehow be described by a unified theory expressed in terms of some generalization of Riemannian geometry. Perhaps particles were some specific singularities or special solutions to the non-linear equations for the metric. The advent of quantum mechanics convinced Weyl (unlike Einstein), that this was a misguided notion, that matter should be described by a complex wave function. The right mathematics was not the geometry of a metric, but (in modern language) the geometry of gauge fields and of sections of a vector bundle with connection. The close connection between the basic ideas of representation theory and of quantum mechanics was quite clear to him, so, unlike Einstein, he enthusiastically adopted the new point of view of quantum physics.

One part of the close connection between Weyl and the history of quantum mechanics isn’t mentioned by Scholz. Weyl was not only a close friend of Schrodinger’s, he was Schrodinger’s wife’s lover. Schrodinger didn’t believe much in monogamy; it’s a well-known story that he discovered the Schrodinger equation while on holiday in the mountains with a girlfriend.

1. The theory of continuous groups and their unitary representations ought to be taught in undergraduate physics courses. It is not that hard and plays an absolutely crucial role in quantum mechanics. For example, the connection between angular momentum in quantum mechanics and the rotation group is more important than the connection with classical angular momentum and yet, in the course I was on, at least, the former was hardly mentioned.

Chris,

Are there any undergraduate level texts available? If so, would you care to share a recommendation?

I mostly agree with the statement about the evidence supporting QM and special relativity – although if I were describing these questions, I would probably separate these two things. (Also, I would use a less mathematical language.)

We have a lot of evidence in favor of special relativity, including classical physics (which has no unitary representations), and a lot of evidence for quantum mechanics (in the nonrelativistic regime). Well, if we unify these two structures, we must introduce these unitary representations of the Poincare group, and this is what quantum field theory dictates. Well, we have a lot of evidence for quantum field theory, that goes beyond special relativity and quantum mechanics separately, too.

I am not sure what (nontrivial) is meant by the unitary representations of the diffeomoprhism group in the second part of the text. If we require the representations to be unitary, it’s because we want to get positive squared norms (positive – and conserved – probabilities) – which is a condition for *physical* states. However, physical states also need to be gauge invariant, and therefore they always form many copies of the trivial, singlet representation of the gauge group – at least of the part of the gauge group that is close to the identity (the part that is connected with the identity, and described by a normalizable wave).

On the other hand, if you consider unphysical states in a formalism “before” you impose the physical constraints, there is no reason for the representations to be unitary. Indeed, the Hilbert spaces including the unphysical states are typically non-unitary representations of various algebras. For example, consider QED with all the time-like and longitudinal polarizations of a photon.

2. The fact that “fundamental” particles/fields look a lot like the vectors of unitary irreducible representations of the Poincare group is an excellent endorsement for our theories of special relativity and quantum mechanics. Indeed, one seems to require little more than this representation theory to build up the whole of QFT.

This is indeed my understanding as well. This is my point in a post which I just sent to sps – let’s see if LM accepts it. Anyway, I’m rather proud of the punchline:

At the most basic level, a quantum theory is defined by a Hilbert space and a unitary time evolution. If the theory has some symmetries, they must be realized as unitary operators acting on this Hilbert space as well. If time translation is included among the symmetries, which is the case for the Poincare algebra (and more subtly for diffeomorphisms), requiring a unitary representation of the symmetry algebra seems to be enough for consistency.

From this viewpoint, there is a 1-1 correspondence between general-covariant quantum theories (GCQT) and unitary representations of the diffeomorphism group on a conventional Hilbert space. Namely, if we have a GCQT, its Hilbert space carries a unitary rep of the diffeomorphism group. And if we have a unitary rep of the diffeomorphism group, the Hilbert space on which it acts can be interpreted as the Hilbert space of some GCQT. Since all unitary quantum irreps of the diffeomorphism group are anomalous, apart from the trivial one, all interesting GCQTs carry anomalous reps of the diffeomorphism group. So rather than being inconsistent, the second type of gauge anomaly is in fact a necessary condition for non-trivial consistency.

1. The theory of continuous groups and their unitary representations ought to be taught in undergraduate physics courses. It is not that hard and plays an absolutely crucial role in quantum mechanics. For example, the connection between angular momentum in quantum mechanics and the rotation group is more important than the connection with classical angular momentum and yet, in the course I was on, at least, the former was hardly mentioned.

2. The fact that “fundamental” particles/fields look a lot like the vectors of unitary irreducible representations of the Poincare group is an excellent endorsement for our theories of special relativity and quantum mechanics. Indeed, one seems to require little more than this representation theory to build up the whole of QFT.

I don’t think it is quite right to say Weyl regarded “pure infinitesimal geometry” as “misguided”. He was, I think astonished that it didn’t work out as he originally thought – and it must have been on his mind all the time, because when the time came for the idea to reemerge in a new context (gauge theory), he was immediately on it. The idea behind PIG is after all incredibly simple – completely localizing the metric. How can it be that the simpler and more natural idea fails? I wish he were alive to see how his idea succeeds in the most remarkable way possible, by bringing matter into full equality with space and time, just as he wished.

When I was a student, my advisor said “Read Weyl. He writes for smart people.” It was the best advice I ever got. Not only are the results intrinsically interesting, one has the feeling on reading Weyl that a great adventure is underway – that the author is not out to pound his vision into your consciousness, but to illuminate the world and its beauties. If only this spirit had become the consensus, instead of the self-indulgent formalism of Bourbaki.